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integers: (#2  t/2) = (x -f- y) (x  y]. But the same problem for
#2 -f- y2 demands 'imaginaries': xz + y2 = (# + 2/V  1)
(x  y V  1). Carrying this up a step in one of many possible
ways open, we might seek to resolve #2 -f y2 + s2 into two
factors of the first degree. Are the positives> negatives, and
imaginaxies sufficient? Or must some new kind of 'number' be
invented to solve the problem? The latter is the case. It was
found that for the new 'numbers' necessary the rules of common
algebra break down in one important particular: it is no longer
true that the order in which 'numbers' are multiplied together is
indifferent; that is, for the new numbers it is not true that a X &
is equal to b x a. More will be said on this when we come to
Hamilton. For the moment we note that the elementary alge-
braic problem of factorization quickly leads us into regions
where complex numbers are inadequate.
How far can we go, what are the most general numbers
possible, if we insist that for these numbers all the familiar laws
of common algebra are to hold? It was proved in the latter part
of the nineteenth century that the complex numbers x + iy,
where xsy are real numbers and i  V  1, are the most
general for which common algebra is true. The real numbers,
we recall, correspond to the distances measured along a fixed
straight line in either direction (positive, negative) from a fixed
point, and the graph of a function /(#), plotted as y  /(#), in
Cartesian geometry, gives us a picture of a function y of a real
variable x. The mathematicians of the seventeenth and eigh-
teenth centuries imagined their functions as being of this kind.
But if the common algebra and its extensions into the calculus
which they applied to their functions are equally applicable to
complex numbers, which include the real numbers as a very
degenerate case, it was but natural that many of the things the
early analysts found were less than half the whole story
possible. In particular the integral calculus presented many
inexplicable anomalies which were cleared up only when the
field of operations was enlarged to its fullest possible extent and
functions of complex variables were introduced by Gauss and
The importance of elliptic functions hi all this vast and